Bernoulli

  • Bernoulli
  • Volume 23, Number 3 (2017), 2028-2057.

Unbiased simulation of stochastic differential equations using parametrix expansions

Patrik Andersson and Arturo Kohatsu-Higa

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we consider an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations with Hölder continuous coefficients. This Monte Carlo method which is based on an Euler scheme with random time steps, can be considered as an infinite dimensional extension of the Multilevel Monte Carlo method for solutions of stochastic differential equations with Hölder continuous coefficients. In particular, we study the properties of the variance of the proposed method. In most cases, the method has infinite variance and therefore we propose an importance sampling method to resolve this issue.

Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 2028-2057.

Dates
Received: September 2014
Revised: September 2015
First available in Project Euclid: 17 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1489737632

Digital Object Identifier
doi:10.3150/16-BEJ803

Mathematical Reviews number (MathSciNet)
MR3624885

Zentralblatt MATH identifier
06714326

Keywords
importance Sampling Monte Carlo method multidimensional diffusion parametrix

Citation

Andersson, Patrik; Kohatsu-Higa, Arturo. Unbiased simulation of stochastic differential equations using parametrix expansions. Bernoulli 23 (2017), no. 3, 2028--2057. doi:10.3150/16-BEJ803. https://projecteuclid.org/euclid.bj/1489737632


Export citation

References

  • [1] Avikainen, R. (2009). On irregular functionals of SDEs and the Euler scheme. Finance Stoch. 13 381–401.
  • [2] Bally, V. and Kohatsu-Higa, A. (2015). A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25 3095–3138.
  • [3] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 93–128.
  • [4] Beskos, A., Papaspiliopoulos, O. and Roberts, G.O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
  • [5] Broadie, M. and Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res. 54 217–231.
  • [6] Chen, B., Oosterlee, C.W. and van der Weide, H. (2012). A low-bias simulation scheme for the SABR stochastic volatility model. Int. J. Theor. Appl. Finance 15 1250016, 37.
  • [7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1955). Higher Transcendental Functions. Vol. III. New York: McGraw-Hill Book Company, Inc.
  • [8] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice-Hall, Inc.
  • [9] Giles, M. (2008). Improved multilevel Monte Carlo convergence using the Milstein scheme. In Monte Carlo and Quasi-Monte Carlo Methods 2006 343–358. Berlin: Springer.
  • [10] Giles, M.B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [11] Giles, M.B. and Szpruch, L. (2013). Antithetic multilevel Monte Carlo estimation for multidimensional SDEs. In Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proc. Math. Stat. 65 367–384. Heidelberg: Springer.
  • [12] Glynn, P.W. (1983). Randomized estimators for time integrals. Technical report, Mathematics Research Center, University of Wisconsin, Madison.
  • [13] Glynn, P.W. and Whitt, W. (1992). The asymptotic efficiency of simulation estimators. Oper. Res. 40 505–520.
  • [14] Gyöngy, I. and Rásonyi, M. (2011). A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic Process. Appl. 121 2189–2200.
  • [15] McLeish, D. (2011). A general method for debiasing a Monte Carlo estimator. Monte Carlo Methods Appl. 17 301–315.
  • [16] Rhee, C. and Glynn, P.W. (2012). A new approach to unbiased estimation for SDE’s. In Simulation Conference (WSC), Proceedings of the 2012 Winter 1–7. Berlin: IEEE.
  • [17] Rhee, C. and Glynn, P.W. (2015). Unbiased estimation with square root convergence for s.d.e. models. Oper. Res. 63 1026–1043.
  • [18] Stroock, D.W. and Varadhan, S.R.S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Berlin: Springer.
  • [19] Tanaka, Y. (2015). On the approximation of the stochastic differential equation via parametrix method. Master’s thesis, Graduate School of Science and Engineering, Ritsumeikan University, Japan.